Question: $P(x)=2x^4-x^3+2x^2-k$ where $k$ is an unknown integer. $P(x)$ divided by $(x+1)$ has a remainder of $2$. What is the value of $k$ ? $k=$
Solution: We can use the polynomial remainder theorem to solve this problem: For a polynomial $p(x)$ and a number $a$, the remainder on division by $x-a$ is $p(a)$. According to the theorem, the remainder when $P(x)$ is divided by $(x+1)$, which can be rewritten as $(x-({-1}))$, is equal to $P({-1})$. We also know that this remainder is equal to $2$. Therefore, $P({-1})=2$. We can use this equality to find $k$. $\begin{aligned} P({-1})&=2 \\\\ 2({-1})^4-({-1})^3+2({-1})^2-k&=2 \\\\ 2\cdot 1-(-1)+2\cdot 1-k&=2 \\\\ 5-k&=2 \\\\ 3&=k \end{aligned}$ In conclusion, $k=3$.